Diffusion Equation Drift Coefficient Estimation Methods Improve

This paper proposed a new way of structuring drift coefficient observations based on the drift coefficient of diffusion process orbit observations-logarithmic incremental method,Through theoretical and simulate analysis, under appropriate conditions, especially for the majority of financial data, the convergence rate and the finite sample properties of drift coefficient estimator based on logarithmic incremental method are more superior than the result based on traditional direct incremental method.In this paper, we first reviewed the fruit of diffusion estimation in theory and applications., then we do the following work about the nonparametric estimation problem in the drift coefficient of diffusion process.In chapter 2, we mainly discussed the ways of nonparametric estimation of diffusion efficient. First we translated the problem of the nonparametric estimation of drift coefficient into the nonparametric regression by Girsanov theorem. For the structure of 1-dimentional function,we applied the kernel estimation on it. Further more we discussed the conclusion about the uniform convergence,consistency and asymptotic normality of it.Then we proposed the logarithmic incremental method based on the direct incremental method. It proved that logarithmic incremental method was more superior than the direct incremental method trough the comparation of the variance of random errors.In chapter 3, we considered several specific models:linear model, nonlinear models and OU-Stochastic volatility model. Respectively we estimated the drift coefficient by logarithmic incremental method and direct incremental method, compared the convergence rate and the finite sample properties of the two methods. It showed that the logarithmic incremental method was indeed superior to the direct incremental method.In chapter 4, Under the condition that the simulate result of the specific model was excellent, we considered the risk-free interest rate process and conducted an empirical analysis of exchange rate data.